# Good intellectual excercises like Newcomb?

What are some good examples of things like the Newcomb paradox (Amazingly accurate predictor makes a prediction about whether you will take box a or box a+b, guaranteed \$1000 in box b, \$1million in box a only if he predicts you will leave box b behind, no tampering after he makes prediction.)?

Ideally I’m looking for things for things that have no agreed upon “right” answer. (I certainly think there is only 1 right answer to Newcomb but it isn’t agreed by everybody.)

Things like Monty Hall which have a clear fixed answer, even if it may be counter-intuative to some and some may be too stubborn or dense to get it even after having it explained, are not what I’m after.

Also the scenario should be simple enough to be able to explain in a chat session with a casino dealer. Anybody got any good examples?

You might like the recreational logic puzzles of Raymond Smullyan. I particularly enjoyed What is the Name of this Book?

Martin Gardner’s Unexpected Hanging https://www.amazon.com/Unexpected-Hanging-Other-Mathematical-Diversions/dp/0226282562 has some good ones - including the Unexpected Hanging itself (https://en.wikipedia.org/wiki/Unexpected_hanging_paradox).

William Poundstone’s Labyrinths of Reason: Paradox, Puzzles, and the Frailty of Knowledge is probably right up your alley, too.

The St. Petersburg Paradox is a good one.

The St Petersburg Paradox, isn’t really a paradox and doesn’t fit the OP. There is an simple answer to it – namely people don’t evaluate gambles like that at there expected value.

Even if you don’t go to expected utility theory or some alternate, there is still some upper bound you could possibly receive, say a billion maybe if playing against another person. So the prizes stop doubling after 30 coin tosses. Now 30 consecutive heads or tails has a chance of a in a billion and might logically be ignored, but setting the price to 1 billion from there on out reduces the expected value of the gamble to something like 30 dollars which, the last I looked, was quite a bit less than infinity.

Maybe if you were playing against the US Treasury the expected value might be 50.

Well perhaps you should rewrite the wikipedia article to make it clear that your solution is the only correct one and explain why the others are incorrect.

The SPP seems to me to fit the OP better than any of the other ideas in this thread so far. Most of Smullyan’s and Gardner’s puzzles have clear and universally agreed on solutions. The SPP does not.

You’re given two envelopes and told that both contain a sum of money and that one contains twice as much money as the other. You cannot determine from examining the envelopes how much either contains.

You’re allowed to pick one of the envelopes at random, open it, and count the money. You’re then offered a chance to exchange the contents of the opened envelope with the contents of the sealed envelope - but you can’t open the sealed envelope unless you commit to the exchange.

Should you swap envelopes?

Logically, the answer is that there’s no reason to swap. You chose the first envelope at random so it could contain the larger or the smaller sum of money with equal probability.

But consider the expected value of switching. You now have X amount of money. The other envelope contains either half as much money or twice as much money - .5X or 2X. As there was an equal chance of you initially picking the larger or smaller amount, the sealed envelope also has an equal chance of containing .5X or 2X.

So if you switch, you have a fifty percent chance that you’re exchanging X for .5X - a loss of .5X. And you have a fifty percent chance that you’re exchanging X for 2X - a gain of X. So overall, you can expect to gain .75X by switching.

To put it in specific terms, suppose you open the first envelope and find ten dollars inside it. You now know the other envelope contains either five dollars or twenty dollars. By switching, you will either have five dollars less or ten dollars more than you have now with an equal chance of both outcomes.

The standard version doesn’t allow you to open the first envelope. The naive probability argument still holds making it superficially seem even more bizarre. But still it’s just the usual thing with people not understanding conditional probability.

Here’s a nice one (I don’t know its name) https://mathoverflow.net/questions/9037/how-is-it-that-you-can-guess-if-one-of-a-pair-of-random-numbers-is-larger-with https://blog.xkcd.com/2010/02/09/math-puzzle/